$\backslash (10-4\backslash )$

The radiator of a steam heating system has a volume of $30$ L and is filled with superheated water vapor at $300$ kPa and $\backslash (200^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ At this moment both the inlet and the exit valves to the radiator are closed. After a while it is observed that the temperature of the steam drops to $\backslash (80^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ as a result of heat transfer to the room air, which is at $\backslash (24^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ Assuming the surroundings to be at $\backslash (0^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ),$ determine $($a$)$ the amount of heat transfer to the room and $($b$)$ the maximum amount of heat that can be supplied to the room if this heat from the radiator is supplied to a heat engine that is driving a heat pump. Assume the heat engine operates between the radiator and the surroundings. Ans: $($a$)78\mathrm{.}09$ kJ $,($b$)278$ kJ